Page 61 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 31

heart solver, can similarly be solved with fully or semi implicit
discretization methods. In practice, as shown by [151], one can
obtain physiologically meaningful results by considering only one
lumped resistance component, such that for example the pressure
drop across the valve becomes
P = RQ.Insuchcase, Eq.(2.22)
can be written

dV
P arterial
P atrial dP
= + − μ (1.20)
dt R 1 R 2 dt
and it can be used to determine the state parameter κ,which for
each valve verifies R i = κ/A 1.5 ,where R 1 corresponds to the arte-
i
rial valve, and R 2 corresponds to the atrial valve. The algorithmic
steps for valve model update then become the following (for each
side of the heart):

Algorithm 1 Valve update algorithm.
1. from the pressure drop across each valve find its opening
phase and its effective opening area
2. from Eq. (1.20) find the state parameter κ
3. find each valve flow rate and send it to its corresponding re-
mote module (arterial Windkessel or atrial module)

Arterial model
The arterial pressure variation can be modeled using a 3-
element Windkessel model [164], which takes as input the arterial
flow and returns the pressure within the artery. The first element
of the model is a peripheral resistance R p , which accounts for the
distal resistance of the circulatory system mainly due to the small
vessels. The compliance C accounts for the elasticity of the arte-
rial walls, whereas the characteristic resistance R c accounts for the
blood mass and for the compliance of the artery proximal to the
valves. Let Q ar (t) be the arterial flow at time t, defined as positive
when exiting the ventricle, P ar (t) be the arterial pressure at time
t and P r be a constant low pressure of reference (for example the
pressure of the remote venous system for the left side circulation).
If the model is a closed whole body circulation system, one needs
to enforce that P r coincides with the corresponding atrial pres-
sure. When the blood flows into the arteries (Q ar (t) >0),during
ejection, the 3-element model is:

dP ar dQ ar R c Q ar P ar − P r
= R c + 1 + − .
dt dt R p C R p C